Cosine of Angle plus Straight Angle/Proof 2
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Theorem
- $\map \cos {x + \pi} = -\cos x$
Proof
| \(\ds \map \cos {x + \pi}\) | \(=\) | \(\ds \map \Re {\map \cos {x + \pi} + i \, \map \sin {x + \pi} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \map \Re {e^{i \paren {x + \pi} } }\) | Euler's Formula | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \Re {e^{i x + i \pi} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \map \Re {e^{i x} e^{i \pi} }\) | Exponential of Sum: Complex Numbers | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \Re {-e^{i x} }\) | Euler's Identity | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\map \Re {e^{i x} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -\map \Re {\cos x + i \cos x}\) | Euler's Formula | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\cos x\) |
$\blacksquare$